LMFDB - Embedded newform 136.4.b.a.33.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(33,136)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("136.33");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 81x^{4} + 222x^{2} + 64 $ x^6 + 81x^4 + 222x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.2
Root			$0.572006i$ of defining polynomial
Character	$\chi$	$=$	136.33
Dual form			136.4.b.a.33.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.49442i q^{3} -18.9829i q^{5} +7.78768i q^{7} +6.80022 q^{9} +O(q^{10})$	$q-4.49442i q^{3} -18.9829i q^{5} +7.78768i q^{7} +6.80022 q^{9} -11.1952i q^{11} -79.5169 q^{13} -85.3171 q^{15} +(-60.9165 + 34.6726i) q^{17} +104.918 q^{19} +35.0011 q^{21} -105.996i q^{23} -235.351 q^{25} -151.912i q^{27} -193.073i q^{29} +258.022i q^{31} -50.3160 q^{33} +147.833 q^{35} -35.4072i q^{37} +357.382i q^{39} -132.430i q^{41} -214.920 q^{43} -129.088i q^{45} +339.385 q^{47} +282.352 q^{49} +(155.833 + 273.784i) q^{51} +124.915 q^{53} -212.518 q^{55} -471.543i q^{57} +735.850 q^{59} -234.813i q^{61} +52.9580i q^{63} +1509.46i q^{65} +489.619 q^{67} -476.388 q^{69} -265.651i q^{71} +193.309i q^{73} +1057.76i q^{75} +87.1849 q^{77} -852.727i q^{79} -499.151 q^{81} -33.8881 q^{83} +(658.186 + 1156.37i) q^{85} -867.752 q^{87} +13.3476 q^{89} -619.252i q^{91} +1159.66 q^{93} -1991.64i q^{95} -861.058i q^{97} -76.1300i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 42 q^{9}+O(q^{10}) $ 6 * q - 42 * q^9	$ 6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100}) $ 6 * q - 42 * q^9 - 72 * q^13 - 24 * q^15 - 126 * q^17 - 24 * q^19 - 204 * q^21 - 114 * q^25 - 228 * q^33 + 408 * q^35 + 192 * q^43 - 72 * q^47 - 18 * q^49 + 456 * q^51 + 924 * q^53 - 456 * q^55 + 1680 * q^59 - 312 * q^67 + 492 * q^69 - 1668 * q^77 - 1614 * q^81 + 1296 * q^83 + 1344 * q^85 + 456 * q^87 + 24 * q^89 + 828 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	4.49442i		−	0.864951i	−0.901646	−	0.432475i	$-0.857640\pi$
$3$		−	4.49442i		−	0.864951i	0.901646	−	0.432475i	$-0.142360\pi$
$4$		0			0
$5$		−	18.9829i		−	1.69788i	−0.528487	−	0.848942i	$-0.677240\pi$
$5$		−	18.9829i		−	1.69788i	0.528487	−	0.848942i	$-0.322760\pi$
$6$		0			0
$7$			7.78768i			0.420495i	0.977648	+	0.210248i	$0.0674270\pi$
$7$			7.78768i			0.420495i	−0.977648	+	0.210248i	$0.932573\pi$
$8$		0			0
$9$	6.80022			0.251860
$10$		0			0
$11$		−	11.1952i		−	0.306863i	−0.988159	−	0.153431i	$-0.950968\pi$
$11$		−	11.1952i		−	0.306863i	0.988159	−	0.153431i	$-0.0490324\pi$
$12$		0			0
$13$	−79.5169			−1.69646			−0.848231	−	0.529626i	$-0.822332\pi$
$13$	−79.5169			−1.69646			−0.848231	+	0.529626i	$0.822332\pi$
$14$		0			0
$15$	−85.3171			−1.46859
$16$		0			0
$17$	−60.9165	+	34.6726i	−0.869083	+	0.494666i
$17$	−60.9165	+	34.6726i	−0.869083	+	0.494666i
$18$		0			0
$19$	104.918			1.26683			0.633414	−	0.773813i	$-0.281653\pi$
$19$	104.918			1.26683			0.633414	+	0.773813i	$0.281653\pi$
$20$		0			0
$21$	35.0011			0.363708
$22$		0			0
$23$		−	105.996i		−	0.960939i	−0.877012	−	0.480469i	$-0.840466\pi$
$23$		−	105.996i		−	0.960939i	0.877012	−	0.480469i	$-0.159534\pi$
$24$		0			0
$25$	−235.351			−1.88281
$26$		0			0
$27$		−	151.912i		−	1.08280i
$28$		0			0
$29$		−	193.073i		−	1.23630i	−0.786058	−	0.618152i	$-0.787882\pi$
$29$		−	193.073i		−	1.23630i	0.786058	−	0.618152i	$-0.212118\pi$
$30$		0			0
$31$			258.022i			1.49491i	0.664314	+	0.747454i	$0.268724\pi$
$31$			258.022i			1.49491i	−0.664314	+	0.747454i	$0.731276\pi$
$32$		0			0
$33$	−50.3160			−0.265421
$34$		0			0
$35$	147.833			0.713952
$36$		0			0
$37$		−	35.4072i		−	0.157322i	−0.996901	−	0.0786609i	$-0.974936\pi$
$37$		−	35.4072i		−	0.157322i	0.996901	−	0.0786609i	$-0.0250644\pi$
$38$		0			0
$39$			357.382i			1.46736i
$40$		0			0
$41$		−	132.430i		−	0.504442i	−0.967670	−	0.252221i	$-0.918839\pi$
$41$		−	132.430i		−	0.504442i	0.967670	−	0.252221i	$-0.0811610\pi$
$42$		0			0
$43$	−214.920			−0.762208			−0.381104	−	0.924532i	$-0.624456\pi$
$43$	−214.920			−0.762208			−0.381104	+	0.924532i	$0.624456\pi$
$44$		0			0
$45$		−	129.088i		−	0.427629i
$46$		0			0
$47$	339.385			1.05328			0.526642	−	0.850087i	$-0.323451\pi$
$47$	339.385			1.05328			0.526642	+	0.850087i	$0.323451\pi$
$48$		0			0
$49$	282.352			0.823184
$50$		0			0
$51$	155.833	+	273.784i	0.427862	+	0.751714i
$52$		0			0
$53$	124.915			0.323744			0.161872	−	0.986812i	$-0.448247\pi$
$53$	124.915			0.323744			0.161872	+	0.986812i	$0.448247\pi$
$54$		0			0
$55$	−212.518			−0.521017
$56$		0			0
$57$		−	471.543i		−	1.09574i
$58$		0			0
$59$	735.850			1.62372			0.811860	−	0.583852i	$-0.198455\pi$
$59$	735.850			1.62372			0.811860	+	0.583852i	$0.198455\pi$
$60$		0			0
$61$		−	234.813i		−	0.492864i	−0.969160	−	0.246432i	$-0.920742\pi$
$61$		−	234.813i		−	0.492864i	0.969160	−	0.246432i	$-0.0792582\pi$
$62$		0			0
$63$			52.9580i			0.105906i
$64$		0			0
$65$			1509.46i			2.88040i
$66$		0			0
$67$	489.619			0.892784			0.446392	−	0.894838i	$-0.352709\pi$
$67$	489.619			0.892784			0.446392	+	0.894838i	$0.352709\pi$
$68$		0			0
$69$	−476.388			−0.831165
$70$		0			0
$71$		−	265.651i		−	0.444042i	−0.975042	−	0.222021i	$-0.928735\pi$
$71$		−	265.651i		−	0.444042i	0.975042	−	0.222021i	$-0.0712653\pi$
$72$		0			0
$73$			193.309i			0.309933i	0.987920	+	0.154966i	$0.0495269\pi$
$73$			193.309i			0.309933i	−0.987920	+	0.154966i	$0.950473\pi$
$74$		0			0
$75$			1057.76i			1.62854i
$76$		0			0
$77$	87.1849			0.129034
$78$		0			0
$79$		−	852.727i		−	1.21442i	−0.794541	−	0.607210i	$-0.792289\pi$
$79$		−	852.727i		−	1.21442i	0.794541	−	0.607210i	$-0.207711\pi$
$80$		0			0
$81$	−499.151			−0.684707
$82$		0			0
$83$	−33.8881			−0.0448158			−0.0224079	−	0.999749i	$-0.507133\pi$
$83$	−33.8881			−0.0448158			−0.0224079	+	0.999749i	$0.507133\pi$
$84$		0			0
$85$	658.186	+	1156.37i	0.839886	+	1.47560i
$86$		0			0
$87$	−867.752			−1.06934
$88$		0			0
$89$	13.3476			0.0158971			0.00794856	−	0.999968i	$-0.497470\pi$
$89$	13.3476			0.0158971			0.00794856	+	0.999968i	$0.497470\pi$
$90$		0			0
$91$		−	619.252i		−	0.713355i
$92$		0			0
$93$	1159.66			1.29302
$94$		0			0
$95$		−	1991.64i		−	2.15093i
$96$		0			0
$97$		−	861.058i		−	0.901311i	−0.892698	−	0.450656i	$-0.851190\pi$
$97$		−	861.058i		−	0.901311i	0.892698	−	0.450656i	$-0.148810\pi$
$98$		0			0
$99$		−	76.1300i		−	0.0772864i
$100$		0			0
$101$	−849.027			−0.836449			−0.418224	−	0.908344i	$-0.637347\pi$
$101$	−849.027			−0.836449			−0.418224	+	0.908344i	$0.637347\pi$
$102$		0			0
$103$	192.677			0.184321			0.0921604	−	0.995744i	$-0.470623\pi$
$103$	192.677			0.184321			0.0921604	+	0.995744i	$0.470623\pi$
$104$		0			0
$105$		−	664.423i		−	0.617533i
$106$		0			0
$107$			1501.60i			1.35669i	0.734745	+	0.678344i	$0.237302\pi$
$107$			1501.60i			1.35669i	−0.734745	+	0.678344i	$0.762698\pi$
$108$		0			0
$109$		−	2206.66i		−	1.93908i	−0.244932	−	0.969540i	$-0.578766\pi$
$109$		−	2206.66i		−	1.93908i	0.244932	−	0.969540i	$-0.421234\pi$
$110$		0			0
$111$	−159.135			−0.136076
$112$		0			0
$113$		−	266.591i		−	0.221936i	−0.993824	−	0.110968i	$-0.964605\pi$
$113$		−	266.591i		−	0.221936i	0.993824	−	0.110968i	$-0.0353951\pi$
$114$		0			0
$115$	−2012.10			−1.63156
$116$		0			0
$117$	−540.732			−0.427271
$118$		0			0
$119$	−270.019	−	474.398i	−0.208005	−	0.365445i
$120$		0			0
$121$	1205.67			0.905835
$122$		0			0
$123$	−595.196			−0.436318
$124$		0			0
$125$			2094.78i			1.49890i
$126$		0			0
$127$	−2484.34			−1.73582			−0.867912	−	0.496718i	$-0.834538\pi$
$127$	−2484.34			−1.73582			−0.867912	+	0.496718i	$0.834538\pi$
$128$		0			0
$129$			965.939i			0.659273i
$130$		0			0
$131$			2471.90i			1.64864i	0.566127	+	0.824318i	$0.308441\pi$
$131$			2471.90i			1.64864i	−0.566127	+	0.824318i	$0.691559\pi$
$132$		0			0
$133$			817.065i			0.532695i
$134$		0			0
$135$	−2883.74			−1.83846
$136$		0			0
$137$	2395.25			1.49372			0.746860	−	0.664981i	$-0.231560\pi$
$137$	2395.25			1.49372			0.746860	+	0.664981i	$0.231560\pi$
$138$		0			0
$139$		−	2383.11i		−	1.45419i	−0.686536	−	0.727096i	$-0.740869\pi$
$139$		−	2383.11i		−	1.45419i	0.686536	−	0.727096i	$-0.259131\pi$
$140$		0			0
$141$		−	1525.34i		−	0.911039i
$142$		0			0
$143$			890.210i			0.520581i
$144$		0			0
$145$	−3665.09			−2.09910
$146$		0			0
$147$		−	1269.01i		−	0.712013i
$148$		0			0
$149$	506.168			0.278302			0.139151	−	0.990271i	$-0.455563\pi$
$149$	506.168			0.278302			0.139151	+	0.990271i	$0.455563\pi$
$150$		0			0
$151$	1073.85			0.578734			0.289367	−	0.957218i	$-0.406555\pi$
$151$	1073.85			0.578734			0.289367	+	0.957218i	$0.406555\pi$
$152$		0			0
$153$	−414.245	+	235.781i	−0.218887	+	0.124587i
$154$		0			0
$155$	4898.01			2.53818
$156$		0			0
$157$	−705.951			−0.358860			−0.179430	−	0.983771i	$-0.557425\pi$
$157$	−705.951			−0.358860			−0.179430	+	0.983771i	$0.557425\pi$
$158$		0			0
$159$		−	561.422i		−	0.280023i
$160$		0			0
$161$	825.459			0.404070
$162$		0			0
$163$			125.718i			0.0604110i	0.999544	+	0.0302055i	$0.00961617\pi$
$163$			125.718i			0.0604110i	−0.999544	+	0.0302055i	$0.990384\pi$
$164$		0			0
$165$			955.144i			0.450654i
$166$		0			0
$167$		−	1578.02i		−	0.731201i	−0.930772	−	0.365601i	$-0.880864\pi$
$167$		−	1578.02i		−	0.731201i	0.930772	−	0.365601i	$-0.119136\pi$
$168$		0			0
$169$	4125.94			1.87799
$170$		0			0
$171$	713.462			0.319063
$172$		0			0
$173$			581.305i			0.255467i	0.991809	+	0.127733i	$0.0407702\pi$
$173$			581.305i			0.255467i	−0.991809	+	0.127733i	$0.959230\pi$
$174$		0			0
$175$		−	1832.84i		−	0.791712i
$176$		0			0
$177$		−	3307.21i		−	1.40444i
$178$		0			0
$179$	681.098			0.284400			0.142200	−	0.989838i	$-0.454582\pi$
$179$	681.098			0.284400			0.142200	+	0.989838i	$0.454582\pi$
$180$		0			0
$181$			2849.49i			1.17017i	0.810971	+	0.585086i	$0.198939\pi$
$181$			2849.49i			1.17017i	−0.810971	+	0.585086i	$0.801061\pi$
$182$		0			0
$183$	−1055.35			−0.426303
$184$		0			0
$185$	−672.132			−0.267114
$186$		0			0
$187$	388.167	+	681.973i	0.151795	+	0.266689i
$188$		0			0
$189$	1183.04			0.455311
$190$		0			0
$191$	−2912.53			−1.10337			−0.551684	−	0.834053i	$-0.686015\pi$
$191$	−2912.53			−1.10337			−0.551684	+	0.834053i	$0.686015\pi$
$192$		0			0
$193$			3800.71i			1.41752i	0.705451	+	0.708758i	$0.250744\pi$
$193$			3800.71i			1.41752i	−0.705451	+	0.708758i	$0.749256\pi$
$194$		0			0
$195$	6784.15			2.49140
$196$		0			0
$197$			1724.83i			0.623801i	0.950115	+	0.311901i	$0.100966\pi$
$197$			1724.83i			0.623801i	−0.950115	+	0.311901i	$0.899034\pi$
$198$		0			0
$199$			5202.68i			1.85331i	0.375916	+	0.926654i	$0.377328\pi$
$199$			5202.68i			1.85331i	−0.375916	+	0.926654i	$0.622672\pi$
$200$		0			0
$201$		−	2200.55i		−	0.772214i
$202$		0			0
$203$	1503.59			0.519860
$204$		0			0
$205$	−2513.91			−0.856484
$206$		0			0
$207$		−	720.793i		−	0.242022i
$208$		0			0
$209$		−	1174.58i		−	0.388742i
$210$		0			0
$211$			2714.47i			0.885647i	0.896609	+	0.442824i	$0.146023\pi$
$211$			2714.47i			0.885647i	−0.896609	+	0.442824i	$0.853977\pi$
$212$		0			0
$213$	−1193.95			−0.384074
$214$		0			0
$215$			4079.80i			1.29414i
$216$		0			0
$217$	−2009.39			−0.628602
$218$		0			0
$219$	868.811			0.268077
$220$		0			0
$221$	4843.89	−	2757.05i	1.47437	−	0.839183i
$222$		0			0
$223$	113.246			0.0340068			0.0170034	−	0.999855i	$-0.494587\pi$
$223$	113.246			0.0340068			0.0170034	+	0.999855i	$0.494587\pi$
$224$		0			0
$225$	−1600.44			−0.474204
$226$		0			0
$227$			5682.46i			1.66149i	0.556653	+	0.830745i	$0.312085\pi$
$227$			5682.46i			1.66149i	−0.556653	+	0.830745i	$0.687915\pi$
$228$		0			0
$229$	68.6647			0.0198144			0.00990718	−	0.999951i	$-0.496846\pi$
$229$	68.6647			0.0198144			0.00990718	+	0.999951i	$0.496846\pi$
$230$		0			0
$231$		−	391.845i		−	0.111608i
$232$		0			0
$233$		−	1367.19i		−	0.384410i	−0.981355	−	0.192205i	$-0.938436\pi$
$233$		−	1367.19i		−	0.384410i	0.981355	−	0.192205i	$-0.0615639\pi$
$234$		0			0
$235$		−	6442.51i		−	1.78835i
$236$		0			0
$237$	−3832.51			−1.05041
$238$		0			0
$239$	5928.60			1.60456			0.802279	−	0.596949i	$-0.203620\pi$
$239$	5928.60			1.60456			0.802279	+	0.596949i	$0.203620\pi$
$240$		0			0
$241$			2340.15i			0.625487i	0.949838	+	0.312744i	$0.101248\pi$
$241$			2340.15i			0.625487i	−0.949838	+	0.312744i	$0.898752\pi$
$242$		0			0
$243$		−	1858.24i		−	0.490560i
$244$		0			0
$245$		−	5359.86i		−	1.39767i
$246$		0			0
$247$	−8342.72			−2.14913
$248$		0			0
$249$			152.307i			0.0387634i
$250$		0			0
$251$	−3402.16			−0.855547			−0.427773	−	0.903886i	$-0.640702\pi$
$251$	−3402.16			−0.855547			−0.427773	+	0.903886i	$0.640702\pi$
$252$		0			0
$253$	−1186.64			−0.294876
$254$		0			0
$255$	5197.22	−	2958.16i	1.27632	−	0.726460i
$256$		0			0
$257$	1106.95			0.268675			0.134338	−	0.990936i	$-0.457109\pi$
$257$	1106.95			0.268675			0.134338	+	0.990936i	$0.457109\pi$
$258$		0			0
$259$	275.740			0.0661531
$260$		0			0
$261$		−	1312.94i		−	0.311376i
$262$		0			0
$263$	5769.53			1.35272			0.676359	−	0.736572i	$-0.263557\pi$
$263$	5769.53			1.35272			0.676359	+	0.736572i	$0.263557\pi$
$264$		0			0
$265$		−	2371.26i		−	0.549680i
$266$		0			0
$267$		−	59.9897i		−	0.0137502i
$268$		0			0
$269$			2722.84i			0.617154i	0.951199	+	0.308577i	$0.0998527\pi$
$269$			2722.84i			0.617154i	−0.951199	+	0.308577i	$0.900147\pi$
$270$		0			0
$271$	−5157.18			−1.15600			−0.578001	−	0.816036i	$-0.696167\pi$
$271$	−5157.18			−1.15600			−0.578001	+	0.816036i	$0.696167\pi$
$272$		0			0
$273$	−2783.18			−0.617017
$274$		0			0
$275$			2634.81i			0.577763i
$276$		0			0
$277$		−	146.678i		−	0.0318159i	−0.999873	−	0.0159080i	$-0.994936\pi$
$277$		−	146.678i		−	0.0318159i	0.999873	−	0.0159080i	$-0.00506387\pi$
$278$		0			0
$279$			1754.61i			0.376507i
$280$		0			0
$281$	−5326.32			−1.13075			−0.565376	−	0.824833i	$-0.691269\pi$
$281$	−5326.32			−1.13075			−0.565376	+	0.824833i	$0.691269\pi$
$282$		0			0
$283$		−	105.637i		−	0.0221888i	−0.999938	−	0.0110944i	$-0.996468\pi$
$283$		−	105.637i		−	0.0221888i	0.999938	−	0.0110944i	$-0.00353154\pi$
$284$		0			0
$285$	−8951.26			−1.86045
$286$		0			0
$287$	1031.32			0.212116
$288$		0			0
$289$	2508.63	−	4224.26i	0.510610	−	0.859812i
$290$		0			0
$291$	−3869.95			−0.779590
$292$		0			0
$293$	−161.544			−0.0322099			−0.0161049	−	0.999870i	$-0.505127\pi$
$293$	−161.544			−0.0322099			−0.0161049	+	0.999870i	$0.505127\pi$
$294$		0			0
$295$		−	13968.6i		−	2.75689i
$296$		0			0
$297$	−1700.69			−0.332270
$298$		0			0
$299$			8428.43i			1.63020i
$300$		0			0
$301$		−	1673.73i		−	0.320505i
$302$		0			0
$303$			3815.88i			0.723487i
$304$		0			0
$305$	−4457.43			−0.836826
$306$		0			0
$307$	1195.49			0.222247			0.111124	−	0.993807i	$-0.464555\pi$
$307$	1195.49			0.222247			0.111124	+	0.993807i	$0.464555\pi$
$308$		0			0
$309$		−	865.971i		−	0.159428i
$310$		0			0
$311$		−	4571.55i		−	0.833533i	−0.909014	−	0.416766i	$-0.863163\pi$
$311$		−	4571.55i		−	0.833533i	0.909014	−	0.416766i	$-0.136837\pi$
$312$		0			0
$313$			9954.23i			1.79759i	0.438367	+	0.898796i	$0.355557\pi$
$313$			9954.23i			1.79759i	−0.438367	+	0.898796i	$0.644443\pi$
$314$		0			0
$315$	1005.30			0.179816
$316$		0			0
$317$		−	7980.79i		−	1.41402i	−0.707201	−	0.707012i	$-0.750042\pi$
$317$		−	7980.79i		−	1.41402i	0.707201	−	0.707012i	$-0.249958\pi$
$318$		0			0
$319$	−2161.50			−0.379376
$320$		0			0
$321$	6748.84			1.17347
$322$		0			0
$323$	−6391.20	+	3637.76i	−1.10098	+	0.626657i
$324$		0			0
$325$	18714.4			3.19411
$326$		0			0
$327$	−9917.65			−1.67721
$328$		0			0
$329$			2643.02i			0.442901i
$330$		0			0
$331$	3905.25			0.648496			0.324248	−	0.945972i	$-0.394889\pi$
$331$	3905.25			0.648496			0.324248	+	0.945972i	$0.394889\pi$
$332$		0			0
$333$		−	240.777i		−	0.0396231i
$334$		0			0
$335$		−	9294.40i		−	1.51584i
$336$		0			0
$337$		−	6993.52i		−	1.13045i	−0.824937	−	0.565224i	$-0.808790\pi$
$337$		−	6993.52i		−	1.13045i	0.824937	−	0.565224i	$-0.191210\pi$
$338$		0			0
$339$	−1198.17			−0.191964
$340$		0			0
$341$	2888.62			0.458731
$342$		0			0
$343$			4870.04i			0.766640i
$344$		0			0
$345$			9043.23i			1.41122i
$346$		0			0
$347$		−	4152.47i		−	0.642409i	−0.947010	−	0.321205i	$-0.895912\pi$
$347$		−	4152.47i		−	0.642409i	0.947010	−	0.321205i	$-0.104088\pi$
$348$		0			0
$349$	−10528.6			−1.61485			−0.807427	−	0.589968i	$-0.799140\pi$
$349$	−10528.6			−1.61485			−0.807427	+	0.589968i	$0.799140\pi$
$350$		0			0
$351$			12079.6i			1.83693i
$352$		0			0
$353$	−7182.88			−1.08302			−0.541510	−	0.840694i	$-0.682147\pi$
$353$	−7182.88			−1.08302			−0.541510	+	0.840694i	$0.682147\pi$
$354$		0			0
$355$	−5042.83			−0.753931
$356$		0			0
$357$	−2132.14	+	1213.58i	−0.316092	+	0.179914i
$358$		0			0
$359$	10999.1			1.61702			0.808509	−	0.588484i	$-0.200275\pi$
$359$	10999.1			1.61702			0.808509	+	0.588484i	$0.200275\pi$
$360$		0			0
$361$	4148.69			0.604854
$362$		0			0
$363$		−	5418.77i		−	0.783503i
$364$		0			0
$365$	3669.56			0.526230
$366$		0			0
$367$		−	4988.07i		−	0.709469i	−0.934967	−	0.354735i	$-0.884571\pi$
$367$		−	4988.07i		−	0.709469i	0.934967	−	0.354735i	$-0.115429\pi$
$368$		0			0
$369$		−	900.554i		−	0.127049i
$370$		0			0
$371$			972.801i			0.136133i
$372$		0			0
$373$	10899.9			1.51307			0.756535	−	0.653954i	$-0.226891\pi$
$373$	10899.9			1.51307			0.756535	+	0.653954i	$0.226891\pi$
$374$		0			0
$375$	9414.82			1.29648
$376$		0			0
$377$			15352.6i			2.09734i
$378$		0			0
$379$			6706.10i			0.908890i	0.890775	+	0.454445i	$0.150162\pi$
$379$			6706.10i			0.908890i	−0.890775	+	0.454445i	$0.849838\pi$
$380$		0			0
$381$			11165.7i			1.50140i
$382$		0			0
$383$	3324.40			0.443522			0.221761	−	0.975101i	$-0.428820\pi$
$383$	3324.40			0.443522			0.221761	+	0.975101i	$0.428820\pi$
$384$		0			0
$385$		−	1655.02i		−	0.219085i
$386$		0			0
$387$	−1461.50			−0.191970
$388$		0			0
$389$	−307.244			−0.0400460			−0.0200230	−	0.999800i	$-0.506374\pi$
$389$	−307.244			−0.0400460			−0.0200230	+	0.999800i	$0.506374\pi$
$390$		0			0
$391$	3675.13	+	6456.87i	0.475344	+	0.835135i
$392$		0			0
$393$	11109.8			1.42599
$394$		0			0
$395$	−16187.2			−2.06194
$396$		0			0
$397$		−	3486.86i		−	0.440808i	−0.975409	−	0.220404i	$-0.929262\pi$
$397$		−	3486.86i		−	0.440808i	0.975409	−	0.220404i	$-0.0707375\pi$
$398$		0			0
$399$	3672.23			0.460755
$400$		0			0
$401$		−	3534.69i		−	0.440185i	−0.975479	−	0.220092i	$-0.929364\pi$
$401$		−	3534.69i		−	0.440185i	0.975479	−	0.220092i	$-0.0706359\pi$
$402$		0			0
$403$		−	20517.1i		−	2.53605i
$404$		0			0
$405$			9475.34i			1.16255i
$406$		0			0
$407$	−396.392			−0.0482762
$408$		0			0
$409$	2689.60			0.325164			0.162582	−	0.986695i	$-0.448018\pi$
$409$	2689.60			0.325164			0.162582	+	0.986695i	$0.448018\pi$
$410$		0			0
$411$		−	10765.2i		−	1.29199i
$412$		0			0
$413$			5730.56i			0.682767i
$414$		0			0
$415$			643.296i			0.0760919i
$416$		0			0
$417$	−10710.7			−1.25780
$418$		0			0
$419$		−	16591.8i		−	1.93451i	−0.253801	−	0.967256i	$-0.581681\pi$
$419$		−	16591.8i		−	1.93451i	0.253801	−	0.967256i	$-0.418319\pi$
$420$		0			0
$421$	9344.11			1.08172			0.540860	−	0.841113i	$-0.318099\pi$
$421$	9344.11			1.08172			0.540860	+	0.841113i	$0.318099\pi$
$422$		0			0
$423$	2307.89			0.265280
$424$		0			0
$425$	14336.7	−	8160.22i	1.63632	−	0.931361i
$426$		0			0
$427$	1828.65			0.207247
$428$		0			0
$429$	4000.97			0.450277
$430$		0			0
$431$			5470.81i			0.611414i	0.952126	+	0.305707i	$0.0988928\pi$
$431$			5470.81i			0.611414i	−0.952126	+	0.305707i	$0.901107\pi$
$432$		0			0
$433$	12423.2			1.37880			0.689402	−	0.724379i	$-0.257874\pi$
$433$	12423.2			1.37880			0.689402	+	0.724379i	$0.257874\pi$
$434$		0			0
$435$			16472.5i			1.81562i
$436$		0			0
$437$		−	11120.8i		−	1.21734i
$438$		0			0
$439$			14332.3i			1.55819i	0.626907	+	0.779094i	$0.284321\pi$
$439$			14332.3i			1.55819i	−0.626907	+	0.779094i	$0.715679\pi$
$440$		0			0
$441$	1920.06			0.207327
$442$		0			0
$443$	9930.54			1.06504			0.532522	−	0.846416i	$-0.321244\pi$
$443$	9930.54			1.06504			0.532522	+	0.846416i	$0.321244\pi$
$444$		0			0
$445$		−	253.376i		−	0.0269914i
$446$		0			0
$447$		−	2274.93i		−	0.240717i
$448$		0			0
$449$			6122.26i			0.643491i	0.946826	+	0.321745i	$0.104269\pi$
$449$			6122.26i			0.643491i	−0.946826	+	0.321745i	$0.895731\pi$
$450$		0			0
$451$	−1482.59			−0.154794
$452$		0			0
$453$		−	4826.34i		−	0.500577i
$454$		0			0
$455$	−11755.2			−1.21119
$456$		0			0
$457$	−12450.1			−1.27438			−0.637192	−	0.770705i	$-0.719904\pi$
$457$	−12450.1			−1.27438			−0.637192	+	0.770705i	$0.719904\pi$
$458$		0			0
$459$	5267.19	+	9253.96i	0.535624	+	0.941041i
$460$		0			0
$461$	−18591.3			−1.87828			−0.939138	−	0.343540i	$-0.888374\pi$
$461$	−18591.3			−1.87828			−0.939138	+	0.343540i	$0.888374\pi$
$462$		0			0
$463$	−7944.33			−0.797418			−0.398709	−	0.917078i	$-0.630542\pi$
$463$	−7944.33			−0.797418			−0.398709	+	0.917078i	$0.630542\pi$
$464$		0			0
$465$		−	22013.7i		−	2.19540i
$466$		0			0
$467$	−4671.36			−0.462880			−0.231440	−	0.972849i	$-0.574344\pi$
$467$	−4671.36			−0.462880			−0.231440	+	0.972849i	$0.574344\pi$
$468$		0			0
$469$			3813.00i			0.375412i
$470$		0			0
$471$			3172.84i			0.310396i
$472$		0			0
$473$			2406.08i			0.233893i
$474$		0			0
$475$	−24692.4			−2.38519
$476$		0			0
$477$	849.452			0.0815382
$478$		0			0
$479$			12306.7i			1.17392i	0.809618	+	0.586958i	$0.199675\pi$
$479$			12306.7i			1.17392i	−0.809618	+	0.586958i	$0.800325\pi$
$480$		0			0
$481$			2815.47i			0.266891i
$482$		0			0
$483$		−	3709.96i		−	0.349501i
$484$		0			0
$485$	−16345.4			−1.53032
$486$		0			0
$487$			14701.7i			1.36796i	0.729498	+	0.683982i	$0.239754\pi$
$487$			14701.7i			1.36796i	−0.729498	+	0.683982i	$0.760246\pi$
$488$		0			0
$489$	565.029			0.0522525
$490$		0			0
$491$	13561.9			1.24652			0.623260	−	0.782015i	$-0.285808\pi$
$491$	13561.9			1.24652			0.623260	+	0.782015i	$0.285808\pi$
$492$		0			0
$493$	6694.35	+	11761.3i	0.611558	+	1.07445i
$494$		0			0
$495$	−1445.17			−0.131223
$496$		0			0
$497$	2068.81			0.186718
$498$		0			0
$499$			5281.77i			0.473837i	0.971530	+	0.236919i	$0.0761375\pi$
$499$			5281.77i			0.473837i	−0.971530	+	0.236919i	$0.923863\pi$
$500$		0			0
$501$	−7092.26			−0.632453
$502$		0			0
$503$		−	8548.70i		−	0.757789i	−0.925440	−	0.378894i	$-0.876304\pi$
$503$		−	8548.70i		−	0.757789i	0.925440	−	0.378894i	$-0.123696\pi$
$504$		0			0
$505$			16117.0i			1.42019i
$506$		0			0
$507$		−	18543.7i		−	1.62437i
$508$		0			0
$509$	6793.69			0.591601			0.295801	−	0.955250i	$-0.404414\pi$
$509$	6793.69			0.591601			0.295801	+	0.955250i	$0.404414\pi$
$510$		0			0
$511$	−1505.43			−0.130325
$512$		0			0
$513$		−	15938.3i		−	1.37172i
$514$		0			0
$515$		−	3657.57i		−	0.312955i
$516$		0			0
$517$		−	3799.49i		−	0.323213i
$518$		0			0
$519$	2612.63			0.220966
$520$		0			0
$521$		−	20578.9i		−	1.73048i	−0.501361	−	0.865238i	$-0.667167\pi$
$521$		−	20578.9i		−	1.73048i	0.501361	−	0.865238i	$-0.332833\pi$
$522$		0			0
$523$	−20180.7			−1.68726			−0.843632	−	0.536921i	$-0.819587\pi$
$523$	−20180.7			−1.68726			−0.843632	+	0.536921i	$0.819587\pi$
$524$		0			0
$525$	−8237.54			−0.684792
$526$		0			0
$527$	−8946.28	−	15717.8i	−0.739480	−	1.29920i
$528$		0			0
$529$	931.953			0.0765968
$530$		0			0
$531$	5003.94			0.408950
$532$		0			0
$533$			10530.4i			0.855767i
$534$		0			0
$535$	28504.8			2.30350
$536$		0			0
$537$		−	3061.14i		−	0.245992i
$538$		0			0
$539$		−	3160.99i		−	0.252604i
$540$		0			0
$541$		−	20689.0i		−	1.64416i	−0.569372	−	0.822080i	$-0.692814\pi$
$541$		−	20689.0i		−	1.64416i	0.569372	−	0.822080i	$-0.307186\pi$
$542$		0			0
$543$	12806.8			1.01214
$544$		0			0
$545$	−41888.8			−3.29233
$546$		0			0
$547$		−	15410.4i		−	1.20457i	−0.798280	−	0.602287i	$-0.794256\pi$
$547$		−	15410.4i		−	1.20457i	0.798280	−	0.602287i	$-0.205744\pi$
$548$		0			0
$549$		−	1596.78i		−	0.124133i
$550$		0			0
$551$		−	20256.8i		−	1.56619i
$552$		0			0
$553$	6640.76			0.510658
$554$		0			0
$555$			3020.84i			0.231041i
$556$		0			0
$557$	−15696.9			−1.19408			−0.597038	−	0.802213i	$-0.703656\pi$
$557$	−15696.9			−1.19408			−0.597038	+	0.802213i	$0.703656\pi$
$558$		0			0
$559$	17089.7			1.29306
$560$		0			0
$561$	3065.07	−	1744.58i	0.230673	−	0.131295i
$562$		0			0
$563$	7928.30			0.593495			0.296748	−	0.954956i	$-0.404098\pi$
$563$	7928.30			0.593495			0.296748	+	0.954956i	$0.404098\pi$
$564$		0			0
$565$	−5060.67			−0.376821
$566$		0			0
$567$		−	3887.23i		−	0.287916i
$568$		0			0
$569$	−2998.46			−0.220918			−0.110459	−	0.993881i	$-0.535232\pi$
$569$	−2998.46			−0.220918			−0.110459	+	0.993881i	$0.535232\pi$
$570$		0			0
$571$		−	173.531i		−	0.0127181i	−0.999980	−	0.00635906i	$-0.997976\pi$
$571$		−	173.531i		−	0.0127181i	0.999980	−	0.00635906i	$-0.00202417\pi$
$572$		0			0
$573$			13090.1i			0.954359i
$574$		0			0
$575$			24946.1i			1.80926i
$576$		0			0
$577$	−7655.22			−0.552324			−0.276162	−	0.961111i	$-0.589063\pi$
$577$	−7655.22			−0.552324			−0.276162	+	0.961111i	$0.589063\pi$
$578$		0			0
$579$	17082.0			1.22608
$580$		0			0
$581$		−	263.910i		−	0.0188448i
$582$		0			0
$583$		−	1398.46i		−	0.0993450i
$584$		0			0
$585$			10264.7i			0.725456i
$586$		0			0
$587$	−12532.9			−0.881242			−0.440621	−	0.897693i	$-0.645242\pi$
$587$	−12532.9			−0.881242			−0.440621	+	0.897693i	$0.645242\pi$
$588$		0			0
$589$			27071.0i			1.89379i
$590$		0			0
$591$	7752.09			0.539558
$592$		0			0
$593$	−704.270			−0.0487705			−0.0243852	−	0.999703i	$-0.507763\pi$
$593$	−704.270			−0.0487705			−0.0243852	+	0.999703i	$0.507763\pi$
$594$		0			0
$595$	−9005.46	+	5125.74i	−0.620484	+	0.353168i
$596$		0			0
$597$	23383.0			1.60302
$598$		0			0
$599$	4705.26			0.320955			0.160477	−	0.987040i	$-0.448697\pi$
$599$	4705.26			0.320955			0.160477	+	0.987040i	$0.448697\pi$
$600$		0			0
$601$			14439.0i			0.979998i	0.871723	+	0.489999i	$0.163003\pi$
$601$			14439.0i			0.979998i	−0.871723	+	0.489999i	$0.836997\pi$
$602$		0			0
$603$	3329.52			0.224857
$604$		0			0
$605$		−	22887.1i		−	1.53800i
$606$		0			0
$607$			11794.6i			0.788678i	0.918965	+	0.394339i	$0.129026\pi$
$607$			11794.6i			0.788678i	−0.918965	+	0.394339i	$0.870974\pi$
$608$		0			0
$609$		−	6757.78i		−	0.449654i
$610$		0			0
$611$	−26986.8			−1.78686
$612$		0			0
$613$	−5789.63			−0.381469			−0.190735	−	0.981642i	$-0.561087\pi$
$613$	−5789.63			−0.381469			−0.190735	+	0.981642i	$0.561087\pi$
$614$		0			0
$615$			11298.6i			0.740816i
$616$		0			0
$617$		−	10770.8i		−	0.702778i	−0.936229	−	0.351389i	$-0.885709\pi$
$617$		−	10770.8i		−	0.702778i	0.936229	−	0.351389i	$-0.114291\pi$
$618$		0			0
$619$			8080.97i			0.524720i	0.964970	+	0.262360i	$0.0845008\pi$
$619$			8080.97i			0.524720i	−0.964970	+	0.262360i	$0.915499\pi$
$620$		0			0
$621$	−16102.0			−1.04050
$622$		0			0
$623$			103.947i			0.00668466i
$624$		0			0
$625$	10346.2			0.662155
$626$		0			0
$627$	−5279.03			−0.336243
$628$		0			0
$629$	1227.66	+	2156.88i	0.0778218	+	0.136726i
$630$		0			0
$631$	−4585.87			−0.289319			−0.144660	−	0.989481i	$-0.546209\pi$
$631$	−4585.87			−0.289319			−0.144660	+	0.989481i	$0.546209\pi$
$632$		0			0
$633$	12199.9			0.766041
$634$		0			0
$635$			47160.0i			2.94723i
$636$		0			0
$637$	−22451.8			−1.39650
$638$		0			0
$639$		−	1806.48i		−	0.111836i
$640$		0			0
$641$			13559.1i			0.835494i	0.908563	+	0.417747i	$0.137180\pi$
$641$			13559.1i			0.835494i	−0.908563	+	0.417747i	$0.862820\pi$
$642$		0			0
$643$		−	6443.91i		−	0.395215i	−0.980281	−	0.197607i	$-0.936683\pi$
$643$		−	6443.91i		−	0.395215i	0.980281	−	0.197607i	$-0.0633171\pi$
$644$		0			0
$645$	18336.3			1.11937
$646$		0			0
$647$	19285.1			1.17183			0.585917	−	0.810371i	$-0.300735\pi$
$647$	19285.1			1.17183			0.585917	+	0.810371i	$0.300735\pi$
$648$		0			0
$649$		−	8238.00i		−	0.498259i
$650$		0			0
$651$			9031.05i			0.543710i
$652$		0			0
$653$		−	31514.1i		−	1.88858i	−0.329120	−	0.944288i	$-0.606752\pi$
$653$		−	31514.1i		−	1.88858i	0.329120	−	0.944288i	$-0.393248\pi$
$654$		0			0
$655$	46923.9			2.79919
$656$		0			0
$657$			1314.54i			0.0780597i
$658$		0			0
$659$	26209.9			1.54930			0.774652	−	0.632388i	$-0.217925\pi$
$659$	26209.9			1.54930			0.774652	+	0.632388i	$0.217925\pi$
$660$		0			0
$661$	30063.5			1.76904			0.884518	−	0.466506i	$-0.154487\pi$
$661$	30063.5			1.76904			0.884518	+	0.466506i	$0.154487\pi$
$662$		0			0
$663$	−12391.3	−	21770.4i	−0.725852	−	1.27525i
$664$		0			0
$665$	15510.3			0.904455
$666$		0			0
$667$	−20464.9			−1.18801
$668$		0			0
$669$		−	508.975i		−	0.0294142i
$670$		0			0
$671$	−2628.78			−0.151242
$672$		0			0
$673$			11767.2i			0.673986i	0.941507	+	0.336993i	$0.109410\pi$
$673$			11767.2i			0.673986i	−0.941507	+	0.336993i	$0.890590\pi$
$674$		0			0
$675$			35752.7i			2.03870i
$676$		0			0
$677$			11567.0i			0.656656i	0.944564	+	0.328328i	$0.106485\pi$
$677$			11567.0i			0.656656i	−0.944564	+	0.328328i	$0.893515\pi$
$678$		0			0
$679$	6705.65			0.378997
$680$		0			0
$681$	25539.3			1.43711
$682$		0			0
$683$		−	2733.68i		−	0.153150i	−0.997064	−	0.0765748i	$-0.975602\pi$
$683$		−	2733.68i		−	0.153150i	0.997064	−	0.0765748i	$-0.0243984\pi$
$684$		0			0
$685$		−	45468.7i		−	2.53616i
$686$		0			0
$687$		−	308.608i		−	0.0171384i
$688$		0			0
$689$	−9932.88			−0.549220
$690$		0			0
$691$		−	26629.2i		−	1.46602i	−0.680217	−	0.733011i	$-0.738114\pi$
$691$		−	26629.2i		−	1.46602i	0.680217	−	0.733011i	$-0.261886\pi$
$692$		0			0
$693$	592.876			0.0324986
$694$		0			0
$695$	−45238.4			−2.46905
$696$		0			0
$697$	4591.69	+	8067.18i	0.249531	+	0.438402i
$698$		0			0
$699$	−6144.72			−0.332496
$700$		0			0
$701$	27802.9			1.49800			0.749002	−	0.662567i	$-0.230533\pi$
$701$	27802.9			1.49800			0.749002	+	0.662567i	$0.230533\pi$
$702$		0			0
$703$		−	3714.84i		−	0.199300i
$704$		0			0
$705$	−28955.3			−1.54684
$706$		0			0
$707$		−	6611.95i		−	0.351723i
$708$		0			0
$709$			24341.1i			1.28935i	0.764456	+	0.644676i	$0.223008\pi$
$709$			24341.1i			1.28935i	−0.764456	+	0.644676i	$0.776992\pi$
$710$		0			0
$711$		−	5798.73i		−	0.305864i
$712$		0			0
$713$	27349.2			1.43651
$714$		0			0
$715$	16898.8			0.883885
$716$		0			0
$717$		−	26645.6i		−	1.38786i
$718$		0			0
$719$		−	4680.26i		−	0.242760i	−0.992606	−	0.121380i	$-0.961268\pi$
$719$		−	4680.26i		−	0.242760i	0.992606	−	0.121380i	$-0.0387320\pi$
$720$		0			0
$721$			1500.51i			0.0775061i
$722$		0			0
$723$	10517.6			0.541016
$724$		0			0
$725$			45440.0i			2.32772i
$726$		0			0
$727$	17376.3			0.886454			0.443227	−	0.896409i	$-0.353834\pi$
$727$	17376.3			0.886454			0.443227	+	0.896409i	$0.353834\pi$
$728$		0			0
$729$	−21828.8			−1.10902
$730$		0			0
$731$	13092.1	−	7451.82i	0.662422	−	0.377039i
$732$		0			0
$733$	−10115.3			−0.509710			−0.254855	−	0.966979i	$-0.582028\pi$
$733$	−10115.3			−0.509710			−0.254855	+	0.966979i	$0.582028\pi$
$734$		0			0
$735$	−24089.5			−1.20892
$736$		0			0
$737$		−	5481.40i		−	0.273962i
$738$		0			0
$739$	667.725			0.0332377			0.0166188	−	0.999862i	$-0.494710\pi$
$739$	667.725			0.0332377			0.0166188	+	0.999862i	$0.494710\pi$
$740$		0			0
$741$			37495.6i			1.85889i
$742$		0			0
$743$			16526.8i			0.816028i	0.912976	+	0.408014i	$0.133779\pi$
$743$			16526.8i			0.816028i	−0.912976	+	0.408014i	$0.866221\pi$
$744$		0			0
$745$		−	9608.55i		−	0.472524i
$746$		0			0
$747$	−230.447			−0.0112873
$748$		0			0
$749$	−11694.0			−0.570481
$750$		0			0
$751$		−	1474.31i		−	0.0716354i	−0.999358	−	0.0358177i	$-0.988596\pi$
$751$		−	1474.31i		−	0.0716354i	0.999358	−	0.0358177i	$-0.0114036\pi$
$752$		0			0
$753$			15290.7i			0.740006i
$754$		0			0
$755$		−	20384.8i		−	0.982623i
$756$		0			0
$757$	1689.69			0.0811267			0.0405634	−	0.999177i	$-0.487085\pi$
$757$	1689.69			0.0811267			0.0405634	+	0.999177i	$0.487085\pi$
$758$		0			0
$759$			5333.27i			0.255053i
$760$		0			0
$761$	−3419.09			−0.162867			−0.0814337	−	0.996679i	$-0.525950\pi$
$761$	−3419.09			−0.162867			−0.0814337	+	0.996679i	$0.525950\pi$
$762$		0			0
$763$	17184.8			0.815375
$764$		0			0
$765$	4475.81	+	7863.58i	0.211534	+	0.371645i
$766$		0			0
$767$	−58512.5			−2.75458
$768$		0			0
$769$	−4668.29			−0.218912			−0.109456	−	0.993992i	$-0.534911\pi$
$769$	−4668.29			−0.218912			−0.109456	+	0.993992i	$0.534911\pi$
$770$		0			0
$771$		−	4975.09i		−	0.232391i
$772$		0			0
$773$	16978.6			0.790011			0.395006	−	0.918679i	$-0.370743\pi$
$773$	16978.6			0.790011			0.395006	+	0.918679i	$0.370743\pi$
$774$		0			0
$775$		−	60725.7i		−	2.81462i
$776$		0			0
$777$		−	1239.29i		−	0.0572192i
$778$		0			0
$779$		−	13894.3i		−	0.639041i
$780$		0			0
$781$	−2974.02			−0.136260
$782$		0			0
$783$	−29330.2			−1.33867
$784$		0			0
$785$			13401.0i			0.609303i
$786$		0			0
$787$			14781.8i			0.669524i	0.942303	+	0.334762i	$0.108656\pi$
$787$			14781.8i			0.669524i	−0.942303	+	0.334762i	$0.891344\pi$
$788$		0			0
$789$		−	25930.7i		−	1.17003i
$790$		0			0
$791$	2076.13			0.0933231
$792$		0			0
$793$			18671.6i			0.836126i
$794$		0			0
$795$	−10657.4			−0.475446
$796$		0			0
$797$	−10849.0			−0.482173			−0.241086	−	0.970504i	$-0.577504\pi$
$797$	−10849.0			−0.482173			−0.241086	+	0.970504i	$0.577504\pi$
$798$		0			0
$799$	−20674.1	+	11767.3i	−0.915391	+	0.521024i
$800$		0			0
$801$	90.7666			0.00400385
$802$		0			0
$803$	2164.14			0.0951068
$804$		0			0
$805$		−	15669.6i		−	0.686064i
$806$		0			0
$807$	12237.6			0.533807
$808$		0			0
$809$		−	23341.5i		−	1.01439i	−0.861831	−	0.507196i	$-0.830682\pi$
$809$		−	23341.5i		−	1.01439i	0.861831	−	0.507196i	$-0.169318\pi$
$810$		0			0
$811$		−	12579.9i		−	0.544686i	−0.962200	−	0.272343i	$-0.912201\pi$
$811$		−	12579.9i		−	0.544686i	0.962200	−	0.272343i	$-0.0877986\pi$
$812$		0			0
$813$			23178.5i			0.999884i
$814$		0			0
$815$	2386.49			0.102571
$816$		0			0
$817$	−22548.9			−0.965587
$818$		0			0
$819$		−	4211.05i		−	0.179666i
$820$		0			0
$821$			19026.8i			0.808818i	0.914578	+	0.404409i	$0.132523\pi$
$821$			19026.8i			0.808818i	−0.914578	+	0.404409i	$0.867477\pi$
$822$		0			0
$823$			15509.4i			0.656892i	0.944523	+	0.328446i	$0.106525\pi$
$823$			15509.4i			0.656892i	−0.944523	+	0.328446i	$0.893475\pi$
$824$		0			0
$825$	11841.9			0.499737
$826$		0			0
$827$		−	45665.8i		−	1.92014i	−0.279757	−	0.960071i	$-0.590254\pi$
$827$		−	45665.8i		−	1.92014i	0.279757	−	0.960071i	$-0.409746\pi$
$828$		0			0
$829$	2210.93			0.0926281			0.0463140	−	0.998927i	$-0.485253\pi$
$829$	2210.93			0.0926281			0.0463140	+	0.998927i	$0.485253\pi$
$830$		0			0
$831$	−659.231			−0.0275192
$832$		0			0
$833$	−17199.9	+	9789.86i	−0.715415	+	0.407201i
$834$		0			0
$835$	−29955.3			−1.24149
$836$		0			0
$837$	39196.7			1.61868
$838$		0			0
$839$			21034.8i			0.865556i	0.901501	+	0.432778i	$0.142467\pi$
$839$			21034.8i			0.865556i	−0.901501	+	0.432778i	$0.857533\pi$
$840$		0			0
$841$	−12888.3			−0.528449
$842$		0			0
$843$			23938.7i			0.978045i
$844$		0			0
$845$		−	78322.3i		−	3.18860i
$846$		0			0
$847$			9389.35i			0.380900i
$848$		0			0
$849$	−474.775			−0.0191923
$850$		0			0
$851$	−3753.00			−0.151177
$852$		0			0
$853$		−	29065.5i		−	1.16669i	−0.812225	−	0.583344i	$-0.801744\pi$
$853$		−	29065.5i		−	1.16669i	0.812225	−	0.583344i	$-0.198256\pi$
$854$		0			0
$855$		−	13543.6i		−	0.541732i
$856$		0			0
$857$			31830.9i			1.26876i	0.773023	+	0.634378i	$0.218744\pi$
$857$			31830.9i			1.26876i	−0.773023	+	0.634378i	$0.781256\pi$
$858$		0			0
$859$	21163.3			0.840609			0.420304	−	0.907383i	$-0.361923\pi$
$859$	21163.3			0.840609			0.420304	+	0.907383i	$0.361923\pi$
$860$		0			0
$861$		−	4635.20i		−	0.183470i
$862$		0			0
$863$	−13999.6			−0.552205			−0.276102	−	0.961128i	$-0.589043\pi$
$863$	−13999.6			−0.552205			−0.276102	+	0.961128i	$0.589043\pi$
$864$		0			0
$865$	11034.9			0.433753
$866$		0			0
$867$	−18985.6	−	11274.8i	−0.743695	−	0.441653i
$868$		0			0
$869$	−9546.47			−0.372660
$870$		0			0
$871$	−38933.0			−1.51457
$872$		0			0
$873$		−	5855.38i		−	0.227004i
$874$		0			0
$875$	−16313.5			−0.630282
$876$		0			0
$877$			28142.0i			1.08357i	0.840518	+	0.541783i	$0.182251\pi$
$877$			28142.0i			1.08357i	−0.840518	+	0.541783i	$0.817749\pi$
$878$		0			0
$879$			726.045i			0.0278600i
$880$		0			0
$881$			22384.3i			0.856010i	0.903776	+	0.428005i	$0.140783\pi$
$881$			22384.3i			0.856010i	−0.903776	+	0.428005i	$0.859217\pi$
$882$		0			0
$883$	−9006.97			−0.343271			−0.171636	−	0.985160i	$-0.554905\pi$
$883$	−9006.97			−0.343271			−0.171636	+	0.985160i	$0.554905\pi$
$884$		0			0
$885$	−62780.6			−2.38457
$886$		0			0
$887$			17019.6i			0.644264i	0.946695	+	0.322132i	$0.104399\pi$
$887$			17019.6i			0.644264i	−0.946695	+	0.322132i	$0.895601\pi$
$888$		0			0
$889$		−	19347.3i		−	0.729906i
$890$		0			0
$891$			5588.11i			0.210111i
$892$		0			0
$893$	35607.4			1.33433
$894$		0			0
$895$		−	12929.2i		−	0.482878i
$896$		0			0
$897$	37880.9			1.41004
$898$		0			0
$899$	49817.2			1.84816
$900$		0			0
$901$	−7609.40	+	4331.13i	−0.281361	+	0.160145i
$902$		0			0
$903$	−7522.43			−0.277221
$904$		0			0
$905$	54091.6			1.98681
$906$		0			0
$907$		−	11818.5i		−	0.432664i	−0.976320	−	0.216332i	$-0.930591\pi$
$907$		−	11818.5i		−	0.432664i	0.976320	−	0.216332i	$-0.0694093\pi$
$908$		0			0
$909$	−5773.57			−0.210668
$910$		0			0
$911$		−	683.382i		−	0.0248534i	−0.999923	−	0.0124267i	$-0.996044\pi$
$911$		−	683.382i		−	0.0248534i	0.999923	−	0.0124267i	$-0.00395564\pi$
$912$		0			0
$913$			379.386i			0.0137523i
$914$		0			0
$915$			20033.6i			0.723813i
$916$		0			0
$917$	−19250.4			−0.693244
$918$		0			0
$919$	10293.5			0.369481			0.184740	−	0.982787i	$-0.440856\pi$
$919$	10293.5			0.369481			0.184740	+	0.982787i	$0.440856\pi$
$920$		0			0
$921$		−	5373.01i		−	0.192233i
$922$		0			0
$923$			21123.7i			0.753301i
$924$		0			0
$925$			8333.11i			0.296207i
$926$		0			0
$927$	1310.25			0.0464230
$928$		0			0
$929$		−	43698.8i		−	1.54329i	−0.636056	−	0.771643i	$-0.719435\pi$
$929$		−	43698.8i		−	1.54329i	0.636056	−	0.771643i	$-0.280565\pi$
$930$		0			0
$931$	29623.7			1.04283
$932$		0			0
$933$	−20546.4			−0.720965
$934$		0			0
$935$	12945.8	−	7368.54i	0.452807	−	0.257729i
$936$		0			0
$937$	−22892.0			−0.798133			−0.399066	−	0.916922i	$-0.630666\pi$
$937$	−22892.0			−0.798133			−0.399066	+	0.916922i	$0.630666\pi$
$938$		0			0
$939$	44738.5			1.55483
$940$		0			0
$941$			16342.7i			0.566160i	0.959096	+	0.283080i	$0.0913562\pi$
$941$			16342.7i			0.566160i	−0.959096	+	0.283080i	$0.908644\pi$
$942$		0			0
$943$	−14037.0			−0.484738
$944$		0			0
$945$		−	22457.6i		−	0.773065i
$946$		0			0
$947$		−	15075.5i		−	0.517303i	−0.965971	−	0.258652i	$-0.916722\pi$
$947$		−	15075.5i		−	0.517303i	0.965971	−	0.258652i	$-0.0832782\pi$
$948$		0			0
$949$		−	15371.3i		−	0.525789i
$950$		0			0
$951$	−35869.0			−1.22306
$952$		0			0
$953$	−3445.89			−0.117128			−0.0585641	−	0.998284i	$-0.518652\pi$
$953$	−3445.89			−0.117128			−0.0585641	+	0.998284i	$0.518652\pi$
$954$		0			0
$955$			55288.3i			1.87339i
$956$		0			0
$957$			9714.68i			0.328141i
$958$		0			0
$959$			18653.4i			0.628102i
$960$		0			0
$961$	−36784.4			−1.23475
$962$		0			0
$963$			10211.2i			0.341695i
$964$		0			0
$965$	72148.4			2.40678
$966$		0			0
$967$	32035.2			1.06534			0.532670	−	0.846323i	$-0.321189\pi$
$967$	32035.2			1.06534			0.532670	+	0.846323i	$0.321189\pi$
$968$		0			0
$969$	16349.6	+	28724.7i	0.542028	+	0.952293i
$970$		0			0
$971$	37545.5			1.24088			0.620439	−	0.784255i	$-0.286954\pi$
$971$	37545.5			1.24088			0.620439	+	0.784255i	$0.286954\pi$
$972$		0			0
$973$	18558.9			0.611481
$974$		0			0
$975$		−	84110.2i		−	2.76275i
$976$		0			0
$977$	−40678.6			−1.33206			−0.666030	−	0.745925i	$-0.732008\pi$
$977$	−40678.6			−1.33206			−0.666030	+	0.745925i	$0.732008\pi$
$978$		0			0
$979$		−	149.429i		−	0.00487823i
$980$		0			0
$981$		−	15005.8i		−	0.488377i
$982$		0			0
$983$		−	9808.59i		−	0.318256i	−0.987258	−	0.159128i	$-0.949132\pi$
$983$		−	9808.59i		−	0.318256i	0.987258	−	0.159128i	$-0.0508682\pi$
$984$		0			0
$985$	32742.3			1.05914
$986$		0			0
$987$	11878.8			0.383088
$988$		0			0
$989$			22780.5i			0.732435i
$990$		0			0
$991$			42497.3i			1.36223i	0.732176	+	0.681115i	$0.238505\pi$
$991$			42497.3i			1.36223i	−0.732176	+	0.681115i	$0.761495\pi$
$992$		0			0
$993$		−	17551.8i		−	0.560917i
$994$		0			0
$995$	98762.0			3.14670
$996$		0			0
$997$			6793.90i			0.215812i	0.994161	+	0.107906i	$0.0344146\pi$
$997$			6793.90i			0.215812i	−0.994161	+	0.107906i	$0.965585\pi$
$998$		0			0
$999$	−5378.79			−0.170348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.b.a.33.2	✓	6
3.2	odd	2		1224.4.c.c.577.6		6
4.3	odd	2		272.4.b.e.33.5		6
17.4	even	4		2312.4.a.h.1.2		6
17.13	even	4		2312.4.a.h.1.5		6
17.16	even	2	inner	136.4.b.a.33.5	yes	6
51.50	odd	2		1224.4.c.c.577.1		6
68.67	odd	2		272.4.b.e.33.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.a.33.2	✓	6	1.1	even	1	trivial
136.4.b.a.33.5	yes	6	17.16	even	2	inner
272.4.b.e.33.2		6	68.67	odd	2
272.4.b.e.33.5		6	4.3	odd	2
1224.4.c.c.577.1		6	51.50	odd	2
1224.4.c.c.577.6		6	3.2	odd	2
2312.4.a.h.1.2		6	17.4	even	4
2312.4.a.h.1.5		6	17.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.49442i q^{3} -18.9829i q^{5} +7.78768i q^{7} +6.80022 q^{9} +O(q^{10})\)	\(q-4.49442i q^{3} -18.9829i q^{5} +7.78768i q^{7} +6.80022 q^{9} -11.1952i q^{11} -79.5169 q^{13} -85.3171 q^{15} +(-60.9165 + 34.6726i) q^{17} +104.918 q^{19} +35.0011 q^{21} -105.996i q^{23} -235.351 q^{25} -151.912i q^{27} -193.073i q^{29} +258.022i q^{31} -50.3160 q^{33} +147.833 q^{35} -35.4072i q^{37} +357.382i q^{39} -132.430i q^{41} -214.920 q^{43} -129.088i q^{45} +339.385 q^{47} +282.352 q^{49} +(155.833 + 273.784i) q^{51} +124.915 q^{53} -212.518 q^{55} -471.543i q^{57} +735.850 q^{59} -234.813i q^{61} +52.9580i q^{63} +1509.46i q^{65} +489.619 q^{67} -476.388 q^{69} -265.651i q^{71} +193.309i q^{73} +1057.76i q^{75} +87.1849 q^{77} -852.727i q^{79} -499.151 q^{81} -33.8881 q^{83} +(658.186 + 1156.37i) q^{85} -867.752 q^{87} +13.3476 q^{89} -619.252i q^{91} +1159.66 q^{93} -1991.64i q^{95} -861.058i q^{97} -76.1300i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 42 q^{9}+O(q^{10}) \) 6 * q - 42 * q^9	\( 6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100}) \) 6 * q - 42 * q^9 - 72 * q^13 - 24 * q^15 - 126 * q^17 - 24 * q^19 - 204 * q^21 - 114 * q^25 - 228 * q^33 + 408 * q^35 + 192 * q^43 - 72 * q^47 - 18 * q^49 + 456 * q^51 + 924 * q^53 - 456 * q^55 + 1680 * q^59 - 312 * q^67 + 492 * q^69 - 1668 * q^77 - 1614 * q^81 + 1296 * q^83 + 1344 * q^85 + 456 * q^87 + 24 * q^89 + 828 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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